Cubature formulas on a sphere invariant under the dihedral group of rotations with inversion $\mathrm{D_{4h}}$

An algorithm of searching for the best (in a sense) cubature formulas on a sphere that are invariant under the dihedral group of rotations with inversion $\mathrm{D_{4h}}$ is described. This algorithm is applied to find parameters of all the best cubature formulas of this symmetry group up to the 29th order of accuracy.